There is a polynomial subring \(^{SU}[\frac{1}{2}] \subset ^W[\frac{1}{2}]\) . This follows from the fact that any \(SU\) -linear multiplication (REF ) induces the standard multiplication on \(^{SU}\) , because
\(^{SU}\subset \mathop {\rm Ker}\partial \) . Furthermore, \(^{SU}[\frac{1}{2}]\)  coincides with \(\mathop {\rm Ker}\partial = \mathop {\rm Im}\partial \)  on \(^W[\frac{1}{2}]\)  (see [1]} or [2]}). From (REF ) we obtain the identity \(x=\frac{1}{2}(\partial ([\mathbb {C}P^1] x) + [\mathbb {C}P^1]\partial x)\)  for any
\(x\in ^W[\frac{1}{2}]\) , which implies that \(^W[\frac{1}{2}]\)  is a free \(^{SU}[\frac{1}{2}]\) -module with basis \(\lbrace 1, [\mathbb {C}P^1]\rbrace \) . Since \([\mathbb {C}P^1] \in A\)  by assumption, we need to show that \(^{SU}[\frac{1}{2}] \subset A[\frac{1}{2}]\) .
By the theorem of Novikov [3]}, \(^{SU}[\frac{1}{2}]\cong \mathbb {Z}[\frac{1}{2}][y_2, y_3, \ldots ]\) , \(\dim y_k=2k\) . The polynomial generators \(y_k\)  are specified by the condition \(s_k(y_k)=\pm m_k m_{k-1}\)  up to a power of 2 (see [1]}).
Denote \(x_1=[\mathbb {C}P^1]\in A\) . We can assume by induction that \(A[\frac{1}{2}]\)  coincides with \(^W[\frac{1}{2}]\)  in dimensions less than \(2k\) . For \(x_k \in A\) , let \(\widetilde{y}_k = \frac{1}{2} \partial (x_1 x_k)=x_k - \frac{1}{2}x_1\partial x_k\) . We have \(\widetilde{y}_k \in ^{SU}[\frac{1}{2}]\) . By the induction hypothesis, \(\partial x_k \in A[\frac{1}{2}]\) , so \(\widetilde{y}_k \in A[\frac{1}{2}]\) . Since \(x_1 \partial x_k\)  is decomposable, \(s_k(\widetilde{y}_k) = s_k(x_k)=m_km_{k-1}\) . It follows that \(\widetilde{y}_k\)  is a polynomial generator of \(^{SU}[\frac{1}{2}]\) . Therefore, we obtain \(^{SU}[\frac{1}{2}] \subset A[\frac{1}{2}]\)  by induction.
